$ 0.\overline{5} \div 0.\overline{1} = {?} $
Solution: First convert the repeating decimals to fractions. $\begin{align*} 10x &= 5.5555...\\ x &= 0.5555...\end{align*} $ $\begin{align*} 9x &= 5 \\ x &= \dfrac{5}{9}\end{align*} $ $\begin{align*} 10y &= 1.1111...\\ y &= 0.1111...\end{align*} $ $\begin{align*} 9y &= 1 \\ y &= \dfrac{1}{9}\end{align*} $ So, the problem becomes: $ \dfrac{5}{9} \div \dfrac{1}{9} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ \dfrac{5}{9} \times \dfrac{9}{1} = {?} $ $ \phantom{\dfrac{5}{9} \times \dfrac{1}{9}} = \dfrac{5 \times 9}{9 \times 1} $ $ \phantom{\dfrac{5}{9} \times \dfrac{1}{9}} = \dfrac{5 \times \cancel{9}} {\cancel{9} \times 1} $ $ \phantom{\dfrac{5}{9} \times \dfrac{1}{9}} = \dfrac{5}{1} $